Retrieve Apple Stock Data
# silence warnings
getSymbols.warning4.0=FALSE
# stock data for Apple and the S&P 500
getSymbols("AAPL", src='yahoo', )
[1] "AAPL"getSymbols("^GSPC", src='yahoo',)
[1] "^GSPC"# Save apple data to a dataframe
df <- data.frame(Date=index(AAPL),coredata(AAPL))
# take most recent observations
df <- tail(df, 4000)
# reset index
rownames(df) = NULL
# SPX to a dataframe
SPX <- data.frame(Date=index(GSPC),coredata(GSPC))
# take most recent observations
SPX <- tail(SPX, 4000)
# reset index
rownames(SPX) = NULL
# copy the data
raw_data = df
# rename AAPL columns for simplicity
names(raw_data)[names(raw_data)=="AAPL.Open"] = "Open"
names(raw_data)[names(raw_data)=="AAPL.High"] = "High"
names(raw_data)[names(raw_data)=="AAPL.Low"] = "Low"
names(raw_data)[names(raw_data)=="AAPL.Close"] = "Close"
names(raw_data)[names(raw_data)=="AAPL.Volume"] = "Volume"
names(raw_data)[names(raw_data)=="AAPL.Adjusted"] = "Price"
# rename SPX columns for simplicity
names(SPX)[names(SPX)=="GSPC.Open"] = "Open"
names(SPX)[names(SPX)=="GSPC.High"] = "High"
names(SPX)[names(SPX)=="GSPC.Low"] = "Low"
names(SPX)[names(SPX)=="GSPC.Close"] = "Close"
names(SPX)[names(SPX)=="GSPC.Volume"] = "Volume"
names(SPX)[names(SPX)=="GSPC.Adjusted"] = "Price"
Sample Period
# retrieve max and min dates
date_max = max(raw_data$Date)
date_min = min(raw_data$Date)
print(paste("Our sample considers AAPL OHLC data from", date_min, "through", date_max))
[1] "Our sample considers AAPL OHLC data from 2007-01-03 through 2021-09-29"Calculate Daily Return
Calculate a simple day-to-day return, from adjusted-close to adjusted-close:
# create a blank column
raw_data$return = NA
raw_data$log_return = NA
# loop through data and calculate daily returns
for(t in 2:nrow(raw_data)){
raw_data$return[t] = (raw_data$Price[t] - raw_data$Price[t-1])/ raw_data$Price[t-1]
raw_data$log_return[t] = log(raw_data$Price[t]/raw_data$Price[t-1])
}
# Do the same for SP500
SPX$return = NA
SPX$log_return = NA
for(t in 2:nrow(SPX)){
SPX$return[t] = (SPX$Price[t] - SPX$Price[t-1])/ SPX$Price[t-1]
SPX$log_return[t] = log(SPX$Price[t]/SPX$Price[t-1])
}
head(SPX, 4)
Date Open High Low Close Volume Price
1 2007-01-03 1418.03 1429.42 1407.86 1416.60 3429160000 1416.60
2 2007-01-04 1416.60 1421.84 1408.43 1418.34 3004460000 1418.34
3 2007-01-05 1418.34 1418.34 1405.75 1409.71 2919400000 1409.71
4 2007-01-08 1409.26 1414.98 1403.97 1412.84 2763340000 1412.84
return log_return
1 NA NA
2 0.001228286 0.001227532
3 -0.006084581 -0.006103168
4 0.002220318 0.002217857head(raw_data,4)
Date Open High Low Close Volume Price
1 2007-01-03 3.081786 3.092143 2.925000 2.992857 1238319600 2.569716
2 2007-01-04 3.001786 3.069643 2.993571 3.059286 847260400 2.626754
3 2007-01-05 3.063214 3.078571 3.014286 3.037500 834741600 2.608047
4 2007-01-08 3.070000 3.090357 3.045714 3.052500 797106800 2.620926
return log_return
1 NA NA
2 0.022196227 0.021953476
3 -0.007121718 -0.007147198
4 0.004938178 0.004926025Indicators and Studies
Simple Moving Averages Functions
Create a function to calculate two simple moving averages of differing lengths for use for a trend-following trading system:
Two_MovAvg_function = function(variable, slow_period = 100, fast_period = 20){
# blank list of variables
fast_ma = rep(NA, length(variable))
slow_ma = rep(NA, length(variable))
# loop to calculate average price for each record
for (t in (slow_period+1):length(variable)){
est_slow = mean(variable[(t-(slow_period+1)):(t-1)])
est_fast = mean(variable[(t-(fast_period+1)):(t-1)])
# update current average price
fast_ma[t] = est_fast
slow_ma[t] = est_slow
}
# create a dataframe for new vectors
ma_data = data.frame(fast_ma = fast_ma,
slow_ma = slow_ma)
return(ma_data)
}
Exponential Moving Averages Function
Create a function to calculate two exponential moving averages of differing lengths for use for a trend-following trading system:
EA_Mov_Avg = function(variable, slow_lag = 100, fast_lag = 25){
fast_ea = variable
slow_ea = variable
ema_diff = rep(NA, length(variable))
# Loop to calculate each exponential average
for (t in 2:length(variable)){
est_slow = slow_ea[t-1] + (slow_ea[t] - slow_ea[t-1]) / ((slow_lag + 1) / 2)
est_fast = fast_ea[t-1] + (fast_ea[t] - fast_ea[t-1]) / ((fast_lag + 1) / 2)
ema_diff[t] = est_slow - est_fast
slow_ea[t] = est_slow
fast_ea[t] = est_fast
}
# dataframe for new vector
ema_data = data.frame(fast_ema = fast_ea,
slow_ema = slow_ea,
ema_diff = ema_diff)
return(ema_data)
}
# add these as new vectors to original data
new_df = EA_Mov_Avg(raw_data$Close, slow_lag = 150, fast_lag = 30)
raw_data$slow_ma = new_df$slow_ema
raw_data$fast_ma = new_df$fast_ema
raw_data$ema_diff = new_df$ema_diff
Average True Range
- The average true range (ATR) is a technical analysis indicator, introduced by market technician J. Welles Wilder Jr. in his book New Concepts in Technical Trading Systems, that measures market volatility by decomposing the entire range of an asset price for that period.
- The true range indicator is taken as the greatest of the following: current high less the current low; the absolute value of the current high less the previous close; and the absolute value of the current low less the previous close. The ATR is then a moving average, generally using 14 days, of the true ranges.
- Here we can use this as a measure of recent volatility in order to adjust our trade entry position sizes accordingly. The idea is that we can enter with smaller position sizes during times of greater volatility and larger positions during times of less volatility in order to decrease overall risk.
True Range
First calculate the maximum of the aforementioned ranges:
max_range_functon = function(variable){
max_range = rep(0, length(variable))
for (t in 2:length(raw_data$Price)){
max_rng = max(abs(raw_data$High[t] - raw_data$Low[t]), abs(raw_data$High[t] - raw_data$Close[t-1]), abs(raw_data$Close[t-1] - raw_data$Low[t]))
max_range[t] = max_rng
}
max_range_data = data.frame(max_range = max_range)
return(max_range_data)
}
# Test Function
# max_range_functon(raw_data$Close[1:10])
# Implement on full dataset
max_range_df = max_range_functon(raw_data$Close)
# Create new variable in data
raw_data$max_range = max_range_df$max_range
head(raw_data)
Date Open High Low Close Volume Price
1 2007-01-03 3.081786 3.092143 2.925000 2.992857 1238319600 2.569716
2 2007-01-04 3.001786 3.069643 2.993571 3.059286 847260400 2.626754
3 2007-01-05 3.063214 3.078571 3.014286 3.037500 834741600 2.608047
4 2007-01-08 3.070000 3.090357 3.045714 3.052500 797106800 2.620926
5 2007-01-09 3.087500 3.320714 3.041071 3.306071 3349298400 2.838647
6 2007-01-10 3.383929 3.492857 3.337500 3.464286 2952880000 2.974494
return log_return slow_ma fast_ma ema_diff max_range
1 NA NA 2.992857 2.992857 NA 0.000000
2 0.022196227 0.021953476 2.993737 2.997143 -0.003405888 0.076786
3 -0.007121718 -0.007147198 2.994316 2.999746 -0.005429937 0.064285
4 0.004938178 0.004926025 2.995087 3.003150 -0.008062751 0.052857
5 0.083070258 0.079799840 2.999206 3.022693 -0.023487057 0.279643
6 0.047856250 0.046746410 3.005366 3.051183 -0.045816917 0.186786Average True Range Function
Create function that uses the true range vector to calculate a moving average of the true range with a user-specified lag period:
ATR_function = function(variable, lag = 15){
atr_list = rep(NA, length(variable))
# Loop to create ATR
for (t in (lag+2):length(variable)){
est_atr = mean(variable[(t-(lag)):t])
atr_list[t] = est_atr
}
atr_data = data.frame(atr = atr_list)
return(atr_data)
}
# Add vector to our data
atr_data = ATR_function(raw_data$max_range)
raw_data$atr = atr_data$atr
Risk per lot
This is a multiple of the ATR for volatility-based position sizing:
risk_multiplier = 5
equity = 100000
raw_data$risk_per_lot = raw_data$atr * risk_multiplier
head(raw_data)
Date Open High Low Close Volume Price
1 2007-01-03 3.081786 3.092143 2.925000 2.992857 1238319600 2.569716
2 2007-01-04 3.001786 3.069643 2.993571 3.059286 847260400 2.626754
3 2007-01-05 3.063214 3.078571 3.014286 3.037500 834741600 2.608047
4 2007-01-08 3.070000 3.090357 3.045714 3.052500 797106800 2.620926
5 2007-01-09 3.087500 3.320714 3.041071 3.306071 3349298400 2.838647
6 2007-01-10 3.383929 3.492857 3.337500 3.464286 2952880000 2.974494
return log_return slow_ma fast_ma ema_diff max_range
1 NA NA 2.992857 2.992857 NA 0.000000
2 0.022196227 0.021953476 2.993737 2.997143 -0.003405888 0.076786
3 -0.007121718 -0.007147198 2.994316 2.999746 -0.005429937 0.064285
4 0.004938178 0.004926025 2.995087 3.003150 -0.008062751 0.052857
5 0.083070258 0.079799840 2.999206 3.022693 -0.023487057 0.279643
6 0.047856250 0.046746410 3.005366 3.051183 -0.045816917 0.186786
atr risk_per_lot
1 NA NA
2 NA NA
3 NA NA
4 NA NA
5 NA NA
6 NA NA# save moving average cross-over dates for chart annotations
h = raw_data[which((raw_data$fast_ma > raw_data$slow_ma) & (raw_data$fast_ma[-1] < raw_data$slow_ma[-1]))+1,]
l = raw_data[which((raw_data$fast_ma < raw_data$slow_ma) & (raw_data$fast_ma[-1] > raw_data$slow_ma[-1]))+1,]
h = na.exclude(h)
h
Date Open High Low Close Volume
276 2008-02-06 4.67250 4.711429 4.348929 4.357143 1573272400
429 2008-09-15 5.07250 5.274643 5.012857 5.012857 920634400
1479 2012-11-14 19.48214 19.551786 19.149286 19.174286 477170400
2168 2015-08-12 28.13250 28.855000 27.407499 28.809999 404870000
2997 2018-11-26 43.56000 43.737499 42.564999 43.654999 179994000
3336 2020-04-02 60.08500 61.287498 59.224998 61.232498 165934000
Price return log_return slow_ma fast_ma ema_diff
276 3.741115 -0.05689513 -0.05857779 5.524325 5.47893 0.04539427
429 4.304120 -0.05760740 -0.05933331 5.865623 5.82371 0.04191335
1479 16.609560 -0.01108855 -0.01115048 21.400512 21.37701 0.02350558
2168 26.453583 0.01541985 0.01530218 30.470711 30.39591 0.07480340
2997 42.441147 0.01352399 0.01343336 50.173922 49.85595 0.31796977
3336 60.567722 0.01668663 0.01654893 66.242964 66.13575 0.10721376
max_range atr risk_per_lot
276 0.362500 0.3355802 1.677901
429 0.306429 0.2209376 1.104688
1479 0.402500 0.6442408 3.221204
2168 1.447501 0.8901563 4.450782
2997 1.172500 1.7617186 8.808593
3336 2.062500 4.4723434 22.361717l
Date Open High Low Close Volume
327 2008-04-21 5.793214 6.017857 5.777143 6.005714 1039152800
576 2009-04-16 4.256786 4.398214 4.242500 4.337500 593446000
1670 2013-08-20 18.203930 18.234644 17.886429 17.895357 358688400
2419 2016-08-10 27.177500 27.225000 26.940001 27.000000 96034000
3081 2019-03-29 47.457500 47.520000 47.134998 47.487499 94256000
3343 2020-04-14 70.000000 72.062500 69.512497 71.762497 194994800
Price return log_return slow_ma fast_ma
327 5.156605 0.044212429 0.043262945 5.224567 5.236748
576 3.724249 0.032387180 0.031873771 3.858784 3.868324
1670 15.798877 -0.013137140 -0.013224196 16.457909 16.526846
2419 25.321693 -0.007444103 -0.007471948 25.482979 25.522523
3081 46.365067 0.006517443 0.006496297 45.244351 45.326908
3343 70.983398 0.050502962 0.049269060 66.259175 66.361778
ema_diff max_range atr risk_per_lot
327 -0.012180766 0.266428 0.1891517 0.9457584
576 -0.009539672 0.196785 0.1381247 0.6906237
1670 -0.068936203 0.348215 0.3600002 1.8000009
2419 -0.039543769 0.284999 0.4454686 2.2273431
3081 -0.082557051 0.385002 1.0609378 5.3046891
3343 -0.102603253 3.750000 3.0849998 15.4249991# Create arrow annotations for trade entry signal
# Upper annotations
h_a <- list(
x = h$Date,
y = h$slow_ma,
text = "Sell",
xref = "x",
yref = "y",
showarrow = TRUE,
arrowcolor = "red",
arrowhead = 5,
arrowsize = 0.8,
arrowwidth = 1.5,
opacity = 0.75,
align = "left",
ax = 5,
ay = -55
)
# Lower annotations
l_a <- list(
x = l$Date,
y = l$slow_ma,
text = "Buy",
xref = "x",
yref = "y",
showarrow = TRUE,
arrowcolor = "green",
arrowhead = 5,
arrowsize = 0.8,
arrowwidth = 1.5,
opacity = 0.75,
align = "right",
ax = -5,
ay = 55
)
# figure labels
f <- list(
family = "Courier New, monospace",
size = 18,
color = "#7f7f7f ")
x <- list(
title = "x Axis",
titlefont = f)
y <- list(
title = "y Axis",
titlefont = f)
Candlestick Chart
Visualizing the Trend-following System
The following candlestick chart helps us to visualize what our strategy is doing
As the faster price smoothing crosses above the slower lag, the system will buy on that day.
# colors column for increasing and decreasing
for (i in 2:length(raw_data[,1])) {
if (raw_data$ema_diff[i] > 0) {
raw_data$MACD_direction[i] = 'Increasing'
} else {
raw_data$MACD_direction[i] = 'Decreasing'
}
}
# color of bars for chart
i <- list(line = list(color = '#17BECF')) # 'i' for increasing
d <- list(line = list(color = '#b22222')) # 'd' for decreasing
# Plot main data
fig <- raw_data %>% plot_ly(x = ~Date, type="ohlc",
open = ~Open, close = ~Close,
high = ~High, low = ~Low, name = "AAPL",
increasing = i, decreasing = d)
# Add Fast and Slow moving average lines
fig <- fig %>% add_lines(x = ~Date, y = ~slow_ma, name = "Slow",
line = list(color = '#ccc', width = 0.5),
legendgroup = "Bands", inherit = F,
showlegend = TRUE, hoverinfo = "none")
fig <- fig %>% add_lines(x = ~Date, y = ~fast_ma, name = "Fast",
line = list(color = '#E377C2', width = 0.5),
hoverinfo = "none", inherit = F)
# Add y-axis title
fig <- fig %>% layout(yaxis = list(title = "Price"))
# plot MACD line subplot
fig2 <- raw_data
fig2 <- fig2 %>% plot_ly(x=~Date, y=~ema_diff, type='bar', name = "MACD",
color = ~MACD_direction, colors = c("red", "green"), alpha = 0.8)
fig2 <- fig2 %>% layout(yaxis = list(title = "MACD"))
# use subplot to add plots to the same figure
fig = subplot(fig, fig2, heights = c(0.8,0.15), nrows = 2,
shareX = TRUE, titleY = TRUE)
# Add arrow annotations
fig <- fig %>% layout(annotations = h_a)
fig <- fig %>% layout(annotations = l_a)
# Add title
fig <- fig %>% layout(title = "AAPL Stock - Moving Average Crossover",
xaxis = list(rangeslider = list(visible = F)))
fig
Generate signals using MA Indicator
Now we use our indicator to generate trading signals and store them in a vector:
raw_data$signal = NA
# Loop through data to create signal column
for(t in 100:nrow(raw_data)){
current_slow = raw_data$slow_ma[t]
current_fast = raw_data$fast_ma[t]
prev_slow = raw_data$slow_ma[t-1]
prev_fast = raw_data$fast_ma[t-1]
# use 'if' statement to translate crossover into signals
if(current_fast > current_slow & prev_fast < prev_slow) {signal = "B"} else
if(current_fast < current_slow & prev_fast > prev_slow) {signal = "S"} else
{signal = "H"}
raw_data$signal[t] = signal
}
Holding Status and Investment Return
After generating the signals, we must calculate the return from our strategy:
# used to calculate fill price penalty / skid
# we will penalize by 40% from the open - high for buy orders
# and 40% from the open - low for sell orders
buy_weight = .40
sell_weight = -.40
# Generate a holding status column
# and investment return column
raw_data$holding = 0 # first record, you are not holding
raw_data$Inv_return = NA # obviously the return will also be 0
# we can begin the loop
for (t in 100:nrow(raw_data)){
if(t==1){prev_holding=0}
else {prev_holding=raw_data$holding[t-1]}
# look at the signal to decide the change of holding status
signal = raw_data$signal[t]
if(signal=="B"){raw_data$holding[t]=1} else
if (signal=="S") {raw_data$holding[t]=0} else
if (signal=="H"){raw_data$holding[t]=prev_holding}
# Now calculate investment return
Open = raw_data$Open[t]
Close = raw_data$Close[t]
High = raw_data$High[t]
Low = raw_data$Low[t]
if(prev_holding==0 & signal=="B"){investment_return=Close/((High-Open)*buy_weight+Open) - 1} else
if(prev_holding==0 & signal=="H"){investment_return=0} else
#if(prev_holding==1 & signal=="S"){investment_return=Close[t-1]/(Open*(1+sell_weight)) - 1} else
#if(prev_holding==0 & signal=="S"){investment_return=(-1)*(Close/(Open*(1+sell_weight)) - 1)} else
if(prev_holding==1 & signal=="B"){investment_return=Close/raw_data$Close[t-1]-1} else
if(prev_holding==1 & signal=="H"){investment_return=Close/raw_data$Close[t-1]-1} else
{investment_return = Open/raw_data$Close[t-1]-1}
# save the investment return to the dataset
raw_data$Inv_return[t] = investment_return}
Cumulative Return
From the individual daily returns, we must create a cash and number of stocks vector to hold each days current figures:
# starting investment $100,000
initial_wealth = 100000
# create vectors to store updates
raw_data$cash = rep(initial_wealth, length(raw_data$Close))
raw_data$n_stock = rep(0, length(raw_data$Close))
# loop through signals and simulate the trades
for (t in 1:length(raw_data$signal)){
if(t==1){prev_cash = initial_wealth; n_stock = 0
} else
{prev_cash = raw_data$cash[t-1];
n_stock = raw_data$n_stock[t-1]}
signal = raw_data$signal[t]
Open = raw_data$Open[t]
Close = raw_data$Close[t]
High = raw_data$High[t]
Low = raw_data$Low[t]
# is.na to deal with rows at beginning with no signal
if(is.na(signal)==TRUE){
prev_cash = raw_data$cash[t-1]
n_stock = raw_data$n_stock[t-1]
} else
if (signal=="B"){
buy_skid = (High-Open) * buy_weight + Open
trans_prc = buy_skid
n_change = floor(prev_cash/trans_prc)
stock_n = n_stock + n_change
cash = prev_cash - n_change * trans_prc
raw_data$cash[t] = cash
raw_data$n_stock[t] = stock_n
} else
if (signal =="S"){
sell_skid = (Open-Low) * sell_weight + Open
trans_prc = sell_skid
n_change = n_stock
stock_n = n_change - n_stock
cash = prev_cash + n_change * trans_prc
raw_data$cash[t] = cash
raw_data$n_stock[t] = stock_n
}else
if (signal == "H") {
raw_data$cash[t] = raw_data$cash[t-1]
raw_data$n_stock[t] = raw_data$n_stock[t-1]
}
}
Final Return
Calculate the cash plus the value of the stock on hand for the last day in the sample:
Date Open High Low Close Volume Price
3712 2021-09-29 142.47 144.45 142.03 142.83 74602000 142.83
return log_return slow_ma fast_ma ema_diff max_range
3712 0.006482968 0.006462044 137.9734 147.2232 -9.249815 2.539993
atr risk_per_lot MACD_direction signal holding Inv_return
3712 3.065625 15.32813 Decreasing H 1 0.006482968
cash n_stock
3712 60.70275 16587final_return = raw_data$n_stock[max(index(raw_data))] * raw_data$Close[max(index(raw_data))] + raw_data$cash[max(index(raw_data))-1]
print(paste("Initial Investment:", sprintf("$ %3.2f", initial_wealth)))
[1] "Initial Investment: $ 100000.00"[1] "Final Return: $ 2369181.95"Visualize Days with Largest Loss and Gain
max_loss
Date Open High Low Close Volume Price
3323 2020-03-16 60.4875 64.77 60 60.5525 322423600 59.89511
return log_return slow_ma fast_ma ema_diff max_range
3323 -0.1286471 -0.1377082 67.15978 72.53165 -5.371863 9.4925
atr risk_per_lot MACD_direction signal holding Inv_return
3323 5.212187 26.06094 Decreasing H 1 -0.1286469
cash n_stock
3323 28.24028 19665max_gain
Date Open High Low Close Volume Price
3322 2020-03-13 66.2225 69.98 63.2375 69.4925 370732000 68.73806
return log_return slow_ma fast_ma ema_diff max_range
3322 0.1198085 0.1131577 67.24847 73.35779 -6.109322 7.922504
atr risk_per_lot MACD_direction signal holding Inv_return
3322 4.774375 23.87188 Decreasing H 1 0.1198083
cash n_stock
3322 28.24028 19665# Create arrow annotations
# Upper annotations
h_a <- list(
x = max_gain$Date,
y = max_gain$Price,
text = "Largest Gain",
xref = "x",
yref = "y",
showarrow = TRUE,
arrowcolor = "green",
arrowhead = 5,
arrowsize = 0.8,
arrowwidth = 2.5,
opacity = 0.75,
align = "right",
ax = 5,
ay = -55
)
# Lower annotations
l_a <- list(
x = max_loss$Date,
y = max_loss$Price,
text = "Largest Loss",
xref = "x",
yref = "y",
showarrow = TRUE,
arrowcolor = "green",
arrowhead = 5,
arrowsize = 0.8,
arrowwidth = 2.5,
opacity = 0.75,
align = "right",
ax = -5,
ay = 55
)
# figure labels
f <- list(
family = "Courier New, monospace",
size = 18,
color = "#7f7f7f ")
x <- list(
title = "x Axis",
titlefont = f)
y <- list(
title = "y Axis",
titlefont = f)
Largest Gain
fig <- max_data %>% plot_ly(x = ~Date, type="ohlc",
open = ~Open, close = ~Close,
high = ~High, low = ~Low, name = "AAPL",
increasing = i, decreasing = d)
# color of bars for chart
i <- list(line = list(color = '#17BECF')) # 'i' for increasing
d <- list(line = list(color = '#b22222')) # 'd' for decreasing
# Add Fast and Slow moving average lines
fig <- fig %>% add_lines(x = ~Date, y = ~slow_ma, name = "Slow",
line = list(color = '#ccc', width = 0.5),
legendgroup = "Bands", inherit = F,
showlegend = TRUE, hoverinfo = "none")
fig <- fig %>% add_lines(x = ~Date, y = ~fast_ma, name = "Fast",
line = list(color = '#E377C2', width = 0.5),
hoverinfo = "none", inherit = F)
# Add y-axis title
fig <- fig %>% layout(yaxis = list(title = "Price"))
# Add arrow annotations
fig <- fig %>% layout(annotations = h_a)
# Add title
fig <- fig %>% layout(title = "AAPL Stock - Strategy Largest 1-Day Gain",
xaxis = list(rangeslider = list(visible = F)))
fig
Largest Loss
fig <- min_data %>% plot_ly(x = ~Date, type="ohlc",
open = ~Open, close = ~Close,
high = ~High, low = ~Low, name = "AAPL",
increasing = i, decreasing = d)
# color of bars for chart
i <- list(line = list(color = '#17BECF')) # 'i' for increasing
d <- list(line = list(color = '#b22222')) # 'd' for decreasing
# Add Fast and Slow moving average lines
fig <- fig %>% add_lines(x = ~Date, y = ~slow_ma, name = "Slow",
line = list(color = '#ccc', width = 0.5),
legendgroup = "Bands", inherit = F,
showlegend = TRUE, hoverinfo = "none")
fig <- fig %>% add_lines(x = ~Date, y = ~fast_ma, name = "Fast",
line = list(color = '#E377C2', width = 0.5),
hoverinfo = "none", inherit = F)
# Add y-axis title
fig <- fig %>% layout(yaxis = list(title = "Price"))
# Add arrow annotations
fig <- fig %>% layout(annotations = l_a)
# Add title
fig <- fig %>% layout(title = "AAPL Stock - Strategy Largest 1-day Loss",
xaxis = list(rangeslider = list(visible = F)))
fig
Max Adverse & Favorable Single-Day Returns
Retrieve the days with the largest single-day loss, gain and average daily return for a simple buy and hold system, as well as for our trend-following system.
Compare the results from the two strategies.
max_daily_return = round((max(raw_data$Inv_return, na.rm = TRUE)*100),4)
max_daily_loss = round((min(raw_data$Inv_return, na.rm = TRUE)*100), 4)
avg_daily_return = round((mean(raw_data$Inv_return, na.rm = TRUE)*100),4)
bh_max_return = round((max(raw_data$return, na.rm = TRUE)*100),4)
bh_max_loss = round((min(raw_data$return, na.rm = TRUE)*100), 4)
bh_avg_return = round((mean(raw_data$return, na.rm = TRUE)*100),4)
print(paste("Largest 1-day return with trend system:", max_daily_return,"%"))
[1] "Largest 1-day return with trend system: 11.9808 %"[1] "Largest 1-day loss with trend system: -12.8647 %"[1] "Average 1-day with trend system return: 0.1004 %"[1] "Largest 1-day return buy and hold strategy: 13.905 %"[1] "Largest 1-day loss with buy and hold strategy: -17.9195 %"[1] "Average 1-day with buy and hold strategy: 0.129 %"Performance Evaluation
Distribution of Returns
Create a histogram of returns for a simple buy and hold strategy, as well as for our trend-following system:
p <- raw_data %>%
ggplot(aes(x=Inv_return)) +
geom_histogram(fill="#69b3a2", color="#e9ecef", alpha=0.7, binwidth = .005) +
coord_cartesian(xlim = c(-0.07, 0.07), ylim = c(0, 500)) +
labs(x = "Investment Returns", y = "Frequency of Returns",
title = "EMA Crossover - Histogram of Investment Returns",
caption = "Trend following system - Slow lag: 100 periods, Fast lag: 20 periods")+
theme_classic()
p
bh = raw_data %>%
ggplot(aes(x=return)) +
geom_histogram(fill="#69b3a2", color="#e9ecef", alpha=0.7, binwidth = .005) +
coord_cartesian(xlim = c(-0.07, 0.07), ylim = c(0, 500)) +
labs(x = "Investment Returns", y = "Frequency of Returns",
title = "Buy / Hold - Histogram of Investment Returns",
caption = "This histogram considers a buy and hold strategy")+
theme_classic()
bh
Mean and Standard Deviation
Retrieve the average daily return and volatility of our trend-following system, as well as a simple buy and hold strategy:
average_return = mean(raw_data$Inv_return, na.rm = TRUE) # average return
volatility = sd(raw_data$Inv_return, na.rm = TRUE) # volatility
print(paste("Average return:", round(average_return,6)))
[1] "Average return: 0.001004"[1] "Volatility: 0.015134"bh_average_return = mean(raw_data$return, na.rm = TRUE) # average return
bh_volatility = sd(raw_data$return, na.rm = TRUE) # volatility
print(paste("Average return:", round(bh_average_return,6)))
[1] "Average return: 0.00129"[1] "Volatility: 0.020302"print(paste("Buy and hold strategy experiences much higher volatility for a small amount of additional return"))
[1] "Buy and hold strategy experiences much higher volatility for a small amount of additional return"Sharpe Ratio
The Sharpe ratio was developed by Nobel laureate William F. Sharpe and is used to help investors understand the return of an investment compared to its risk.12 The ratio is the average return earned in excess of the risk-free rate per unit of volatility or total risk. Volatility is a measure of the price fluctuations of an asset or portfolio.
- Our Sharpe ratios are based on daily returns for both the buy and hold strategy and the trend-following system.
rf = .00001 # risk-free rate
print(paste("Trend following: Sharpe ratio of daily returns:",round((average_return-rf)/volatility,5)))
[1] "Trend following: Sharpe ratio of daily returns: 0.0657"print(paste("Buy & hold: Sharpe ratio of daily returns:",round((bh_average_return-rf)/bh_volatility,5)))
[1] "Buy & hold: Sharpe ratio of daily returns: 0.06304"[1] "Our Trend-following system achieves slightly higher Sharpe ratio"Value at Risk
VAR measures the cut-off return that your financial asset will fall below with certain probability.
Value at risk (VaR) is a statistic that quantifies the extent of possible financial losses within a firm, portfolio, or position over a specific time frame. This metric is most commonly used by investmentand commercial banks to determine the extent and probabilities of potential losses in their institutional portfolios.
var = quantile(raw_data$Inv_return, 0.05, na.rm = TRUE)*100
bh_var = quantile(raw_data$return, 0.05, na.rm = TRUE)*100
print(paste("Trend-following VAR: we can expect 95% of the time returns will be greater than:", round(var,4)))
[1] "Trend-following VAR: we can expect 95% of the time returns will be greater than: -2.361"print(paste("Buy / Hold model VAR: we can expect 95% of the time returns will be greater than:", round(bh_var,4)))
[1] "Buy / Hold model VAR: we can expect 95% of the time returns will be greater than: -2.9629"Summary of Investment & Market Return
In order to break down risk into two components, we need to collect what part of the return was attributed to the market and what part to our trading system.
Fit CAPM Model
Our \(y\) is the excess return of the trading system
Our \(x\) is the excess return of the market
y = raw_data$Inv_return - rf
x = SPX$return - rf
Call:
lm(formula = y ~ x)
Residuals:
Min 1Q Median 3Q Max
-0.084232 -0.006302 -0.000674 0.006462 0.099507
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.0007829 0.0002192 3.572 0.00036 ***
x 0.5729293 0.0168314 34.039 < 2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.01317 on 3611 degrees of freedom
(99 observations deleted due to missingness)
Multiple R-squared: 0.2429, Adjusted R-squared: 0.2427
F-statistic: 1159 on 1 and 3611 DF, p-value: < 2.2e-16[1] "Trend-following system Beta: 0.5729"The larger the \(BETA\) is, the larger the systematic risk is.
We use this to measure the aggressiveness of the stock or the trading system.
Our trend-following system is mostly conservative.
Jensen’s Alpha (Abnormal Return)
In the regression model, it is the intercept term
This tells us how much the strategy gives us extra reward, above the market return.
Generally we would like to see a positive \(ALPHA\)
This is associated with its statistical significance.
The higher \(ALPHA\) the better the investment performance
tf_beta = capm$coefficients[2] # beta
jensens_alpha = capm$coefficients[1] # Jensen's Alpha
print(paste("Jensen's Alpha:", round(jensens_alpha,4)))
[1] "Jensen's Alpha: 8e-04"Treynor’s Ratio
[1] "Treynor's Ratio Version 1: 0.0014"print(paste("Treynor's Ratio Version 2:", round(((mean(raw_data$Inv_return, na.rm = TRUE))/tf_beta),4)))
[1] "Treynor's Ratio Version 2: 0.0018"